On phi-Dedekind rings and phi-Krull rings

Abstract

The purpose of this paper is to introduce two new classes of rings that are closely related to the classes of Dedekind domains and Krull domains. Let H = {R | R is a commutative ring with 1 and Nil(R) is a divided prime ideal of R}. Let R in H, T(R) be the total quotient ring of R, and let phi be the map from R into RNil(R) (the localization of R at Nil(R)) such that phi(a/b) = a/b for every a in R and b in R\ Z(R). Then phi is a ring homomorphism from T(R) into RNil(R), and phi restricted to R is also a ring homomorphism from R into RNil(R) given by phi(x) = x /1 for every x in R. A nonnil ideal I of R is said to be phi-invertible if phi(I) is an invertible ideal of phi(R). If every nonnil ideal of R is phi-invertible, then we say that R is a phi-Dedekind ring. Also, we say that R is a phi-Krull ring if phi(R) is the intersection of {Vi}, where each Vi is a discrete phi-chained overring of phi(R), and for every nonnilpotent element x in R , phi(x) is a unit in all but finitely many Vi. We show that the theories of phi-Dedekind and phi-Krull rings resemble those of Dedekind and Krull domains.